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14
Chapter
Percentage
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Percentages
Converting fractions to
percentages
Converting percentages to
fractions
Converting decimals to
percentages
Converting percentages to
decimals
Plotting numbers on a number line
Shaded regions of figures
A
B
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Contents:
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IB MYP_1
260
PERCENTAGE (Chapter 14)
OPENING PROBLEMS
Problem 1:
Mahari has a collection of blue and red
beads. She wants to string them together to
form a bracelet. Can you write the number
of red beads compared with the total number
of beads as:
² a fraction
² a percentage?
² a decimal
Problem 2:
² If a store advertises a 25% off sale, what
percentage of the normal cost of an item
would you have to pay?
² Can you write the amount you would have
to pay as a fraction of the usual amount?
A
PERCENTAGES
From the chapter on fractions, you might remember the difficulty of comparing some fractions.
Fractions with the same denominators like 15 , 35 , 45 and 85 were easy to compare but fractions
3
7
with different denominators like 14 , 10
, 25
or 37
20 needed to first be converted to fractions
with the same denominator.
Percentages are special kinds of fractions because their denominator is always 100.
Rather than write the fraction
100% =
100
100
12
100 ,
we would write 12%.
= 1, so 100% represents the whole amount.
The word percent comes
from the Latin meaning
out of every hundred.
Percentages are comparisons of a portion with the whole
amount, which we call 100%.
For example, 12% =
12
100 ,
and means ‘12 out of every 100’.
EXERCISE 14A
1 In each of the following patterns there are 100 tiles. For each pattern:
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i write the number of coloured tiles as a fraction of the total, leaving your answer
with the denominator 100
ii write a percentage which shows the proportion of squares shaded.
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IB MYP_1
PERCENTAGE (Chapter 14)
a
b
261
c
2 In this circle there are 100 symbols. For
Check that your
each of the different symbols present:
numerators total 100.
X M V
a count how many there are
V M X V C
X X C L X
X
C L V C
X C X
b write the proportion as a
X
X V M X
V M
C X C
V X V X
V
fraction of 100
V
X M V
LM X C
M X V
X V CX V L
X C
L C
X
X
M
c write the proportion as a
C V X V
L CL VV
VM
X C X
XC X
percentage.
X V L V X LV X V
X
C
V L M X V
X C XM
V X V
3 For the numbers from 1 to 100 inclusive, write as a percentage the proportion which:
a are odd
b are exactly divisible by 5
c are multiples of 4
d can be divided by 10 exactly
e contain the digit 1
f have only 1 digit
g are prime numbers
h are composite numbers.
ACTIVITY 1
CATCHING ATTENTION WITH PERCENTAGE
Here are some examples of eye-catching graphs which use percentages to
create an impact.
lemon
squash¡/
lemonade
15%
mineral water
8%
other
17%
food 36%
paper 21%
glass 16%
cola
brands
60%
plastic 10%
garden 7%
steel 5%
other 4%
aluminium 1%
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IB MYP_1
262
PERCENTAGE (Chapter 14)
What to do:
Think of some eye-catching ways you could present different types of information in
percentage form. Remember when you represent a percentage, you need to give a symbol
or statement which explains what the whole quantity is.
ACTIVITY 2
EVERYDAY USE OF PERCENTAGE
What to do:
1 Read the following everyday examples of the use of percentages:
² In my street 25% of the homes have roses growing in the front garden.
² Sixty five percent of students at my school voted for a greater variety of fresh
fruit in the school canteen.
² Twenty seven percent of primary school age children do not eat fruit and
vegetables.
² Our netball goal shooter Alice had a 68% accuracy rate for the whole season.
² Sarah improved by 10% in her times table tests.
² Our country’s unemployment rate dropped to 8:1%.
² Last year over 52% of 5-14 year old children living in Switzerland played sport
outside school hours.
² House prices near the beach increased by 15% in the last year.
² Nearly 27% of the population visited a
museum in 2008.
² The number of children attending the local
cinema during the school holidays has dropped
12% on last year’s attendance.
² The humidity at 9 am was 46% and at 3 pm
it was 88%.
² After the weekend rainfalls the reserviour was
at 75% capacity.
2 For each of the above examples, suggest how and
why these percentages may have been worked out.
3 What is a census? How is a census conducted?
Why is a census conducted? What types of
questions may be asked? Why are percentages
important here?
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4 What census do schools conduct, and why?
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IB MYP_1
PERCENTAGE (Chapter 14)
B
263
CONVERTING FRACTIONS
TO PERCENTAGES
If an object is divided into 100 equal parts then each part is 1 percent and is written as 1%.
1
100
Thus
= 1%
100
100
and
= 100%
Most common fractions and decimal fractions can be changed into percentage form by first
converting into an equal fraction with a denominator of 100.
For example:
The shaded part of both squares is the same.
In the first square
=
1
5
is shaded.
In the second square
1
5
So,
=
20
100
20
100
is shaded.
= 20%:
EXERCISE 14B
1 What percentage is represented by the following shaded diagrams?
a
b
c
d
2 Estimate the percentage shaded:
a
0
3
b
10 20 30 40 50 60 70 80 90 100
0
c
10 20 30 40 50 60 70 80 90 100
0
10 20 30 40 50 60 70 80 90 100
a
Copy and complete:
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100
=
%
IB MYP_1
264
PERCENTAGE (Chapter 14)
b
Copy and complete:
=
=
=
4
%
c
Copy and complete:
=
=
=
Example 1
%
Self Tutor
Write as percentages:
a
19
100
19
100
a
76:8
100
b
c
76:8
100
b
= 19%
557
1000
c
= 76:8%
=
=
557
1000
557¥10
1000¥10
55:7
100
= 55:7%
4 Write the following fractions as percentages:
a
e
i
31
100
79
100
6:6
100
3
100
50
100
34:5
100
b
f
j
37
100
100
100
75
1000
c
g
k
d
h
l
Example 2
54
100
85
100
356
1000
Self Tutor
Write as percentages:
a
2
5
b
2
5
2£20
5£20
40
100
a
=
=
13
25
b
=
=
= 40%
13
25
13£4
25£4
52
100
= 52%
5 Write the following as fractions with denominator 100, and then convert to percentages:
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IB MYP_1
PERCENTAGE (Chapter 14)
6 Write these statements in full:
a Fourteen percent means fourteen out of every .......
b If 53% of the students in a school are girls, 53% means the fraction
265
:::::::
:
:::::::
7 Refer to the illustration given
and then complete the table
which follows:
Students
Number Fraction Fraction with Percentage
denom. 100
a wearing shorts
b with a ball
c wearing skirts
d wearing shorts and with a ball
e wearing track pants, baseball cap
and green top
f
wearing shorts or track pants
g every student in the picture
Example 3
Self Tutor
In a class of 25 students, 6 have black hair.
What percentage of the class have black hair?
The fraction with black hair =
=
=
6
25
6£4
25£4
24
100
So, 24% of the class has black hair.
8 In a class of 25 students, 13 have blue eyes. What percentage of the class have blue
eyes?
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9 There are 35 basketball players in the Tigers club. 14 of them are boys. What percentage
are girls?
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IB MYP_1
266
PERCENTAGE (Chapter 14)
A pack of 52 playing cards has
CARDS
been shuffled. You can view the
whole pack by clicking on the icon.
Suppose the 25 cards shown are
dealt from the pack.
a What percentage of the cards shown are:
i hearts
ii black
iii picture cards
iv spades?
b If an ace is 1 and picture cards are higher than
10, what percentage of the cards shown are:
ii 5 or lower
i 10 or higher
iii higher than 5 and less than 10?
c In the full pack of cards, what percentage are:
i red
ii diamonds
iii either spades or clubs?
10
C
CONVERTING PERCENTAGES
TO FRACTIONS
Percentages are easily converted into fractions. We first write the percentage as a fraction
with a denominator of 100, and then express the fraction in its lowest terms.
Example 4
Self Tutor
Convert to a fraction with
denominator 100, then
write in simplest form.
Express as fractions in lowest terms:
a 70%
b 85%
70%
a
=
=
=
85%
b
70
100
70¥10
100¥10
7
10
=
=
=
85
100
85¥5
100¥5
17
20
EXERCISE 14C
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50%
40%
100%
37%
32%
125%
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1 Write as a fraction in lowest terms:
a 43%
b 37%
e 90%
f 20%
i 75%
j 95%
m 5%
n 44%
q 99%
r 21%
u 200%
v 350%
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30%
25%
3%
80%
15%
800%
IB MYP_1
PERCENTAGE (Chapter 14)
Example 5
267
Self Tutor
Express 2:5% as a fraction
in lowest terms.
2:5%
=
=
=
=
=
2:5
100
2:5£10
100£10
25
1000
25¥25
1000¥25
1
40
2 Write as a fraction in lowest terms:
a 12:5%
b 7:5%
e 97:5%
f 0:2%
D
fto remove the decimalg
c 0:5%
g 0:05%
d 17:3%
h 0:02%
CONVERTING DECIMALS TO
PERCENTAGES
To write a decimal number as a percentage we multiply it by 100%.
Since 100% = 100
100 = 1, multiplying by 100% is the same as multiplying by 1. We
therefore do not change the value of the number.
Example 6
Self Tutor
Remember that
100% = 1.
Convert to a percentage by multiplying by 100%:
a 0:27
a
b 0:055
0:27
= 0:27 £ 100%
= 27%
0:055
= 0:055 £ 100%
= 5:5%
b
Another way of converting a fraction to a percentage is to first convert it to a decimal.
Example 7
Self Tutor
Change to percentages by multiplying by 100%:
4
5
4
5
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= 75%
0
= 80%
5
= 0:75 £ 100%
95
= 0:8 £ 100%
100
= 0:75
50
= 0:8
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IB MYP_1
268
PERCENTAGE (Chapter 14)
EXERCISE 14D
1 Convert into percentage form by multiplying by 100%:
a 0:37
e 0:73
b 0:89
f 0:05
c 0:15
g 1:02
d 0:49
h 1:17
2 Convert into percentage form by multiplying by 100%:
a 0:2
e 0:074
b 0:7
f 0:739
c 0:9
g 0:0067
d 0:4
h 0:0018
3 Convert to a percentage by first writing as a decimal:
a
e
i
m
q
1
10
2
5
19
20
3
8
1
3
8
10
1
2
3
50
b
f
j
c
g
k
n 1
o
4
10
3
20
39
50
11
100
d
1
3
2
3
3
3
d
h
l
p
3
5
1
4
17
25
7
8
2
3
r
4 Copy and complete these patterns:
a 1 is 100%
1
5
2
5
3
5
4
5
5
5
b
1
2 is 50%
1
4 is::::::
1
8 is::::::
1
16 is::::::
E
= 20%
c
= ::::::
= ::::::
is 33 13 %
is ::::::
is ::::::
= ::::::
1
4
2
4
3
4
4
4
is ::::::
=
1
2
is :::::::
= ::::::
= ::::::
= ::::::
CONVERTING PERCENTAGES
TO DECIMALS
To write a percentage as a decimal number, we divide by 100%.
To achieve this we can first write the percentage as a common fraction with denominator 100.
Example 8
Self Tutor
Write as a decimal:
a 21%
b 12 12 %
21%
a
=
=
To divide by
100, move the
decimal point two
places to the left.
12 12 %
b
21
100
21:
100
= 12:5%
=
= 0:21
=
12:5
100
12:5
100
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IB MYP_1
PERCENTAGE (Chapter 14)
269
It is worthwhile remembering the conversions in the following table:
Percentage
100%
Common
Fraction
1
Decimal
Fraction
1:0
Percentage
Common
Fraction
5%
3
4
1
2
1
4
1
5
1
10
0:75
33 13 %
0:5
66 23 %
12 12 %
6 14 %
1
2%
1
20
1
3
2
3
1
8
1
16
1
200
75%
50%
25%
20%
10%
0:25
0:2
0:1
Decimal
Fraction
0:05
0:3
0:6
0:125
0:0625
0:005
EXERCISE 14E
1 Write as a decimal:
a 50%
e 85%
i 15%
b 30%
f 5%
j 100%
c 25%
g 45%
k 67%
d 60%
h 42%
l 125%
2 Write as a decimal:
a 7:5%
e 0:15%
b 18:3%
f 8:63%
c 17:2%
g 37 12 %
d 106:7%
h 6 12 %
i
1
2%
j 1 12 %
3
4%
k
l 4 14 %
3 Copy and complete the table below:
Fraction
a
Percent
20%
b
40%
2
5
Decimal
0:2
Percent
g
0:5
100%
j
0:85
e
3
20
k
2
25
f
5
8
i
3
4
d
0:375
l
a Write 45% as a fraction and as a decimal.
The fraction must be in simplest form.
4
b Write
c Write
7
25 as a decimal and as a
1
5 % as a decimal number
Decimal
0:35
12:5%
h
c
Fraction
percentage.
You must be able to
convert from one
form to another.
and as a fraction.
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d Write 250% as a decimal and as a fraction.
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IB MYP_1
270
PERCENTAGE (Chapter 14)
F
PLOTTING NUMBERS
ON A NUMBER LINE
Plotting numbers on a number line can be difficult, especially when the numbers are given
as a mixture of fractions, decimals, and percentages. However, we can make the comparison
easier by converting all fractions and decimals to percentages.
Example 9
Convert
1
4
²
1
4,
Self Tutor
0:42, and 33% to percentages and plot them on a number line.
£ 100% = 25%
² 0:42 £ 100% = 42%
² 33% is already a percentage
We use the percentages to arrange the numbers in order from lowest to highest.
Qr_
0%
10
20
33%
30
0.42
40
50
60
70
80
90
100%
EXERCISE 14F
1 Convert each set of numbers to percentages and plot them on a number line:
a
3
5,
70%, 0:65
b 55%,
d 0:85, 34 , 92%
g
3
4,
27
50 ,
e
0:65, 42%
9
20 ,
0:83
c 0:93, 79%,
67%, 0:59
f 47%, 0:74,
h 0:39, 58%,
7 2
20 , 5
i
5
8,
73%,
17
20
18
30
13
20 ,
0:47
2 Write each of the following number line positions as fractions with denominator 100, as
decimals, and also as percentages:
a
0%
20
40
60
80
25%
100%
b
0%
20
40
60
80
100%
0%
20
40
60
80
100%
Aha! 25% is
bigger than 12 .
c
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3 Comment on the cartoon opposite.
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IB MYP_1
PERCENTAGE (Chapter 14)
G
271
SHADED REGIONS OF FIGURES
When we shade regions of figures to illustrate percentages, it is important that the region is
the correct size.
In some cases we may divide the figure into a number of equal parts, and then shade the
appropriate number of them.
Example 10
Self Tutor
For the given figure:
a what fraction of the figure is unshaded
b what percentage of the figure is unshaded?
a There are 50 squares in total.
30 squares are unshaded.
3
is unshaded.
So, 30
50 = 5
3
5
£ 100% = 60%
So, 60% is unshaded.
b
When we divide up a circle, we need to remember there
are 360o in a full turn.
80%
Suppose we wish to shade 20% of a circle.
If 100% is 360o then 1% is 3:6o and so 20% is 72o .
72°
20%
EXERCISE 14G
1 Copy and complete the following table, filling in the shading where necessary:
Figure
Fraction shaded
Percentage shaded
Percentage unshaded
a
PRINTABLE
WORKSHEET
3
4
b
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IB MYP_1